The Quadratic Time-Varying Hausdorff and Large Deviation Multifractal Spectrum of Stochastic Fractal Signal

Although multifractal describes the spectrum distribution of Singularity Exponent (SE), it loses the temporal information, and it´s hard to describe the dynamics evolving process of non-stationary system. The time-varying singularity distribution indicates the spatial dynamics character of system. Therefore, the time-varying quadratic multifractal spectrum is proposed. Similar to the Wigner-Ville time-frequency analysis, the time-delayed conjugation of analyzed signal is selected as the windows function, and the quadratic time-singularity exponent distribution of the instantaneous self-correlation is deduced based on the short-time multifractal analysis, i.e. quadratic time-singularity multifractal distribution, which includes Hausdorff Measure, time-varying singular spectrum distribution, time-varying large deviation multifractal spectrum, which exhibits the singular exponent distribution of signal at arbitrary time.